Fusion, the Coulomb barrier, and tunnelling

Fusing light nuclei releases energy because binding energy per nucleon climbs steeply, from 1.11 MeV in ²H to 7.07 MeV in ⁴He. The reaction behind every tokamak design is ²H + ³H → ⁴He + n. With atomic masses the two electrons of the reactant atoms match the two in helium, so Q = (2.014102 + 3.016049 − 4.002603 − 1.008665) u × 931.494 MeV u⁻¹ = 17.6 MeV. The products leave with equal and opposite momenta, so the kinetic energy divides inversely with mass: 14.1 MeV to the neutron, 3.5 MeV to the alpha particle. The Sun starts instead with p + p → ²H + e⁺ + νₑ, worth only 0.42 MeV, in which one proton must become a neutron by the weak interaction during the encounter. Its chain ends at 4 ¹H → ⁴He + 2e⁺ + 2νₑ, 26.7 MeV per helium nucleus once the positrons annihilate. Every one of these is exothermic by MeV. Energy is not what holds fusion back.

Outside contact, two nuclei feel only their Coulomb repulsion, V(r) = Z₁Z₂(1.440 MeV fm)/r. The strong force reaches a femtometre or two, so it switches on only when the surfaces touch, at r ≈ R₁ + R₂ with R = 1.2A¹⁄³ fm, and the potential then drops into a well tens of MeV deep. The barrier therefore peaks at contact: V_C = 1.440/3.24 = 0.44 MeV for D-T and 1.440/2.40 = 0.60 MeV for p-p, but 2.5 MeV for p + ¹⁴N, since the height grows as Z₁Z₂ and falls only as A¹⁄³. The energy that counts is the centre-of-mass kinetic energy E: the motion of the centre of mass is conserved and cannot be spent against the barrier. Classically a pair turns back at r_c = Z₁Z₂(1.440 MeV fm)/E. Two protons at 6.1 keV stop 237 fm apart, a hundred times the contact distance, and that 237 fm of forbidden region is what any quantum rescue has to cross.

At the centre of the Sun, T = 1.57 × 10⁷ K, so kT = 1.35 keV and the mean relative kinetic energy (3/2)kT is 2.0 keV, 300 times below the p-p barrier. A D-T plasma at 1.5 × 10⁸ K, extreme by any terrestrial standard, has kT = 12.9 keV, still 34 times below 0.44 MeV. Pairs are not all average. Relative energies follow the normalised Maxwell–Boltzmann distribution f(E) = (2/√π)(kT)⁻³⁄² √E e^(−E/kT), with ∫₀^∞ f dE = 1, and for V_C ≫ kT the fraction of pairs above the barrier is about (2/√π)√(V_C/kT) e^(−V_C/kT). For the Sun, V_C/kT = 444 and that fraction is 2 × 10⁻¹⁹². Even if all of the Sun's 10⁵⁷ protons sat at core temperature, the expected number above the barrier at any instant would be about 2 × 10⁻¹³⁵. Raising the mean energy to the barrier instead would take 4.6 × 10⁹ K. On classical physics the Sun cannot shine.

Inside the forbidden region the relative wavefunction decays at the local rate κ(r) = √(2mᵣ(V(r) − E))/ℏ, with mᵣ = m₁m₂/(m₁ + m₂) the reduced mass, and the transmission probability is P ≈ exp(−2∫κ dr), taken from contact out to r_c. With V = Er_c/r, and contact far inside the turning point, the lower limit can be set to zero: the integral of √(r_c/r − 1) from 0 to r_c is πr_c/2. The exponent becomes πr_c√(2mᵣE)/ℏ, which rearranges to √(E_G/E) with E_G = 2mᵣ c²(παZ₁Z₂)². For p-p, E_G = 493 keV, so at 6.1 keV P ≈ e^(−9.0) = 1.2 × 10⁻⁴: small, not zero. A size check agrees. At that energy ℏ/p = 83 fm, and the barrier is 237 fm thick, under three reduced wavelengths. Correspondence holds as well: E_G ∝ 1/ℏ², so as ℏ → 0 the exponent diverges for every E below the barrier and the classical verdict returns.

The rate per pair is σv averaged over the thermal distribution. Factor the cross section as σ(E) = (S(E)/E) exp(−√(E_G/E)): 1/E carries the geometric area πλ̄², the exponential carries the barrier, and S(E) carries the nuclear physics, slowly varying for a non-resonant reaction. Then ⟨σv⟩ ∝ ∫ S(E) exp(−E/kT − √(E_G/E)) dE, one exponential falling with E and the other rising. Setting the derivative of the exponent to zero, 1/kT = ½E_G¹⁄²E⁻³⁄², gives E₀ = (E_G(kT)²/4)¹⁄³, and at E₀ the exponent equals exactly 3E₀/kT. For solar p-p, E₀ = 6.1 keV: 4.5 kT and 1% of the barrier, inside a window of 1/e full width 4√(E₀kT/3) = 6.6 keV. The suppression is e^(−13.5) = 1.4 × 10⁻⁶ instead of e⁽⁻⁴⁴⁴⁾, a gain of 10¹⁸⁷. Because 3E₀/kT falls as T⁻¹⁄³, and the prefactor goes as T⁻²⁄³, the rate climbs locally as T^ν with ν = (3E₀/kT − 2)/3 = 3.8.

The barrier and the tail fix where reactions happen and how steeply the rate climbs with temperature. They do not fix how many: that is S(E), and this survey cannot predict it. For D-T, S is about 10⁴ keV b, and a ⁵He resonance lifts the cross section to about 5 b near 64 keV, a feature no smooth S can produce. For p-p, S is about 4 × 10⁻²² keV b, some 10²⁵ times smaller, because a proton must turn into a neutron by the weak interaction while the pair overlaps. That is why a proton in the solar core waits about 10¹⁰ years to fuse, and why the Sun lasts. Smaller effects sharpen the estimate without overturning it: finite nuclear size shortens the tunnel, cutting the p-p exponent by the fraction (4/π)√((R₁ + R₂)/r_c) = 13% at 6.1 keV, and electron screening raises solar rates by a few per cent. Burning is a further question again: a reactor needs enough density and confinement time for fusion power to beat its losses, the Lawson condition.